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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Boolean algebras with no rigid or homogeneous factors


Author: Petr Štěpánek
Journal: Trans. Amer. Math. Soc. 270 (1982), 131-147
MSC: Primary 06E05; Secondary 03E40
MathSciNet review: 642333
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Abstract: A simple construction of Boolean algebras with no rigid or homogeneous factors is described. It is shown that for every uncountable cardinal $ \kappa $ there are $ {2^\kappa }$ isomorphism types of Boolean algebras of power $ \kappa $ with no rigid or homogeneous factors. A similar result is obtained for complete Boolean algebras for certain regular cardinals. It is shown that every Boolean algebra can be completely embedded in a complete Boolean algebra with no rigid or homogeneous factors in such a way that the automorphism group of the smaller algebra is a subgroup of the automorphism group of the larger algebra. It turns out that the cardinalities of antichains in both algebras are the same. It is also shown that every $ \kappa $-distributive complete Boolean algebra can be completely embedded in a $ \kappa $-distributive complete Boolean algebra with no rigid or homogeneous factors.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1982-0642333-3
PII: S 0002-9947(1982)0642333-3
Keywords: Boolean algebras with no rigid or homogeneous factors, automorphisms, embeddings and isomorphism types of BA's, chain conditions, distributivity
Article copyright: © Copyright 1982 American Mathematical Society